Complete Multipartite Graphs of Non-QE Class

We derive a formula for the QE constant of a complete multipartite graph and determine the complete multipartite graphs of non-QE class, namely, those which do not admit quadratic embeddings in a Euclidean space. Moreover, the primary non-QE graphs are specified among the complete multipartite graphs.


Introduction
Realization of a graph G = (V, E) in a Euclidean space R N is of fundamental interest.In this paper, we focus on a particular realization called quadratic embedding, which traces back to the early works of Schoenberg [20,21] and has been studied along with Euclidean distance geometry, see e.g., [1,3,6,7,10].
Let G = (V, E) be a graph, which is always assumed to be finite and connected.A map x, y ∈ V, www.ejgta.org Complete multipartite graphs of non-QE class | Nobuaki Obata where the left-hand side is the square of the Euclidean distance and the right-hand side is the graph distance.A graph G is called of QE class or of non-QE class according as it admits a quadratic embedding or not.It follows from Schoenberg [20,21] that a graph G admits a quadratic embedding if and only if the distance matrix D = [d G (x, y)] is conditionally negative definite, i.e., ⟨f, Df ⟩ ≤ 0 for all real column vectors f indexed by V with ⟨1, f ⟩ = 0, where 1 denotes the column vector of which entries are all one and ⟨•, •⟩ is the canonical inner product.In this connection, a new numeric invariant of a graph was proposed in the recent papers [16,18].The quadratic embedding constant (QE constant for short) of a graph G is defined by where the right-hand side stands for the conditional maximum of the quadratic function ⟨f, Df ⟩ with f running over a unit sphere determined by ⟨f, f ⟩ = 1 and ⟨1, f ⟩ = 0.By definition, a graph G is of QE class if and only if QEC(G) ≤ 0. An advantage of the QE constant is that (1) is determined explicitly or estimated finely by means of the method of Lagrange's multipliers.The QE constants are explicitly known for some special series of graphs, see also [4,5,12,14,15,17,19].
There are interesting questions both on graphs of QE class and on those of non-QE class.One of the important questions on non-QE graphs is to obtain a sufficiently rich list of non-QE graphs.The main purpose of this paper is to determine all complete multipartite graphs of non-QE class.In this paper we first derive a general formula for the QE constant of a complete multipartite graph. ( where α * is the minimal solution to the equation Using the above explicit formula, we determine all complete multipartite graphs with positive QE constants, that is, of non-QE class.
and only if one of the following conditions is satisfied: If a graph H is isometrically embedded in a graph G, we have QEC(H) ≤ QEC(G).Hence, if G contains a non-QE subgraph H isometrically, then G is of non-QE class too.Thus, upon classifying graphs of non-QE class it is important to specify a primary non-QE graph, that is, a non-QE graph G which does not contain a non-QE graph H as an isometrically embedded proper subgraph.
By Theorem 1.2 there are four families of complete multipartite graphs of non-QE class.From each family a primary one is specified as follows.
Theorem 1.3 (Theorem 4.2).Among the complete multipartite graphs there are four primary non-QE graphs which are listed with their QE constants as follows: Moreover, any complete multipartite graph of non-QE class contains at least one of the above four primary non-QE graphs as an isometrically embedded subgraph.
This paper is organized as follows.In Section 2 we prepare basic notions and notations, for more details see [15,18].In Section 3 we prove the main formula for the QE constants of complete multipartite graphs (Theorem 1.1) and show some examples.In Section 4 we determine all complete multipartite graphs of non-QE class (Theorem 1.2) and specify primary ones (Theorem 1.3).All primary non-QE graphs on six or fewer vertices are already determined [17].As a result, we find three new primary non-QE graphs on seven vertices.
The QE constant is not only useful for judging whether nor not a graph G is of QE-class but also interesting as a possible scale of classifying graphs [4].Moreover, the QE constant is interesting from the point of view of spectral analysis of distance matrices, for distance spectra see e.g., [2,8,9].In fact, QEC(G) lies between the largest and the second largest eigenvalues of the distance matrix D in such a way that λ 2 (D) ≤ QEC(G) < λ 1 (D).In this line, an interesting question is to characterize graphs such that λ 2 (D) = QEC(G).The work is now in progress.

Distance matrices
A graph G = (V, E) is a pair of a non-empty set V of vertices and a set E of edges, where V is assumed to be a finite set throughout the paper.For x, y ∈ V we write x ∼ y if {x, y} ∈ E. A graph is called connected if any pair of vertices x, y ∈ V there exists a finite sequence of vertices In that case the sequence of vertices is called a walk from x to y of length m.Unless otherwise stated, by a graph we mean a finite connected graph throughout this paper.
Let G = (V, E) be a graph.For x, y ∈ V with x ̸ = y let d G (x, y) denote the length of a shortest walk connecting x and y.By definition we set d G (x, x) = 0. Then d G (x, y) becomes a metric on V , which we call the graph distance.The distance matrix of G is defined by which is a matrix with index set V × V .For notational simplicity we sometimes write d(x, y) for d G (x, y) when there is no danger of confusion.
Let C(V ) be the linear space of R-valued functions f on V .We always identify The distance matrix D induces a linear transformation on C(V ) by matrix multiplication as usual.

Quadratic embedding
where the left-hand side is the square of the Euclidean distance.A graph G is called of QE class or of non-QE class according as it admits a quadratic embedding or not.
A graph By our convention both G and H are assumed to be connected and hence admit graph distances of their own.We say that H is isometrically embedded in G if In that case we write H → G.
Proof.Obvious by definition.
We say that a graph of non-QE class is primary if it contains no isometrically embedded proper subgraphs of non-QE class.In view of Lemma 2.1, for classifying graphs of non-QE class it is essential to explore primary non-QE graphs.
We mention a simple and useful criterion for isometric embedding.Recall that the diameter of a graph G = (V, E) is defined by Proof.
(1) By assumption d H (x, y) = d G (x, y) for all x, y ∈ V ′ .In particular, for x, y ∈ V ′ d H (x, y) = 1 if and only if d G (x, y)=1.Therefore, if two vertices x, y ∈ V ′ are adjacent in G, so are in H.This means that H is an induced subgraph of G.
(2) We will prove that then x and y are adjacent in G, so are in H since H is an induced subgraph of G.We then obtain d H (x, y) = 1, which is again contradiction.Therefore, we have d G (x, y) = 2 and d H (x, y) = d G (x, y) holds.

Quadratic embedding constants
Let G = (V, E) be a graph with |V | ≥ 2. The quadratic embedding constant (QE constant for short) of G is defined by It follows from Schoenberg [20,21] that a graph G admits a quadratic embedding if and only if the distance matrix D = [d G (x, y)] is conditionally negative definite, i.e., ⟨f, Df ⟩ ≤ 0 for all f ∈ C(V ) with ⟨1, f ⟩ = 0. Hence a graph G is of QE class (resp. of non-QE class) if and only if QEC(G) ≤ 0 (resp.QEC(G) > 0).The idea of QE constant was proposed in [18], where a standard method of computing the QE constants is established on the basis of Lagrange's multipliers.
As the distance matrix of H becomes a principal submatrix of the distance matrix of G, the proof is straightforward, see also [18].Note also that Lemma 2.1 follows immediately from Lemma 2.3.
For later reference we recall some concrete values of the QE constants.Further examples are found in [5,12,15,16,19].Example 2.1 ( [18]).For the complete graph K n with n ≥ 2 we have Conversely, any graph G with QEC(G) = −1 is necessarily a complete graph [4].Moreover, for any graph G on two or more vertices we have QEC(G) ≥ −1.

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Complete multipartite graphs of non-QE class | Nobuaki Obata Example 2.2 ( [18]).For the cycles on odd number of vertices we have and for those on even number of vertices we have Example 2.3 ( [14]).For the paths P n with n ≥ 2 we have Example 2.4 ( [18]).For the complete bipartite graph K m,n with m ≥ 1 and n ≥ 1 we have The above formula will be generalized to the complete multipartite graphs in Subsection 3.2, see Theorem 3.1.
Example 2.5 ([12]).For m ≥ 1 and n ≥ 1 the graph join Km + K n is called the complete split graph.We have For a more general result on the graph join of regular graphs, see [12].
Example 2.6.A table of the QE constants of graphs on n ≤ 5 vertices is available [18].We see directly from the table that all graphs on n ≤ 4 vertices are of QE-class, and that there are 21 graphs on five vertices among which two are of non-QE class.Those graphs are shown in Figure 1, where Gn-x stands for the graph on n vertices with number x in the list of small graphs due to McKay [13].Their QE constants are given by Both are primary non-QE graphs since all graphs on four vertices are of QE class.Note that G5-10 is the complete bipartite graph K 3,2 .

Definition and basic properties
For k ≥ 2 let V 1 , V 2 , . . .V k be mutually disjoint non-empty finite sets.Setting we obtain a graph G = (V, E), which is called a complete k-partite graph with parts V 1 , V 2 , . . ., V k and is denoted by For k ≥ 2 a complete k-partite graph is determined (up to graph isomorphisms) by the sequence m 1 , m 2 , . . ., m k defined by m i = |V i | and is denoted by Without loss of generality we may assume that For simplicity we write K 1(k) for K 1,1,...,1 ("1" appears k times).Obviously, which is nothing else but the complete graph on k vertices.Overusing our notation for the case of k = 1 we understand that KM (V 1 ) is the empty graph on V 1 , that is the complement to the complete graph K m 1 so that We note that (2) is not valid for k = 1.
For later use we list some basic properties of complete multipartite graphs, of which verification is straightforward and is omitted.
. ., V k be mutually disjoint non-empty finite sets.Consider an arbitrary partition: , where p ≥ 1, q ≥ 1 and p + q = k.Then we have , where the right-hand side is the graph join.
For example, we have where (3) is taken into account.
Next we study a subgraph of . This isometric embedding is stated as follows.
Obviously, any induced subgraph of as mentioned above.With the help of Lemma 2.2 we come to the following Lemma 3.4.Let H be a subgraph of a complete k-partite graph K m 1 ,m 2 ,...,m k .If H is a graph on two or more vertices and is isometrically embedded in K m 1 ,m 2 ,...,m k , then H is a complete p-partite graph K n 1 ,n 2 ,...,np of the form as in Lemma 3.3.

Calculating QE constants
Throughout this subsection, letting k ≥ 2 and Our goal is to obtain an explicit formula for QEC(G).Since 1 ≤ diam(G) ≤ 2 the following general result is useful.
Let A be the adjacency matrix of G and set Then we have QEC(G) = −2 − α min .Now let A be the adjacency matrix of G = K m 1 ,m 2 ,...,m k .Then A is written in a block matrix form: where J is a matrix of which entries are all one and the size is understood in the context.For example, in the last block matrix in (5) J's appear as diagonal entries.We understand naturally that their sizes are m 1 × m 1 , . . ., m k × m k from left top to right bottom.We will find the conditional minimum α min in (4).According to the block matrix form of the adjacency matrix A in (5), any Then we have Upon applying the method of Lagrange's multiplier we set By general theory we know that the conditional minimum ( 4) is attained at a certain stationary point.Let S be the set of all stationary points (f, α, β), that is, the set of solutions to After simple calculus (7) becomes the following system of equations: For 1 ≤ s ≤ m i the sth entry of the left-hand side of ( 8) is given by Thus ( 8) is equivalent to As a result, S is the set of all solutions (f = [f i ], α, β) to the equations ( 11), ( 9) and (10).For convenience we denote by S A the set of all α ∈ R appearing in S, that is, (f, α, β) ∈ S for some f and β.

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Complete multipartite graphs of non-QE class | Nobuaki Obata Lemma 3.5.For any (f, α, β) ∈ S we have ⟨f, Af ⟩ = α.Therefore, Noting that the equations ( 8)-( 10) are fulfilled, we have On the other hand, by (10) we have Consequently, we see from ( 6) that as desired.The assertion in the second half is then apparent.
In order to study S A we come back to the equations ( 8)- (10), where ( 8) is equivalent to (11).
We are now in a position to sum up the above argument.(1) By Lemma 3.8 we know that −m 1 ∈ S A .We then see from (22) that min S A = −m 1 so that α min = −m 1 .
We now come to the first main result.Proof.Straightforward from Propositions 3.1 and 3.2.